Abstract
Given a k-coloring of a graph G, a Kempe-change for two colors a and b produces another k-coloring of G, as follows: first choose a connected component in the subgraph of G induced by the two color classes of a and b, and then swap the colors a and b in the component. Two k-colorings are called Kempe-equivalent if one can be transformed into the other by a sequence of Kempe-changes. We consider two problems, defined as follows: First, given two k-colorings of a graph G, Kempe Reachability asks whether they are Kempe-equivalent; and second, given a graph G and a positive integer k, Kempe Connectivity asks whether any two k-colorings of G are Kempe-equivalent. We analyze the complexity of these problems from the viewpoint of graph classes. We prove that Kempe Reachability is PSPACE-complete for any fixed \(k \ge 3\), and that it remains PSPACE-complete even when restricted to three colors and planar graphs of maximum degree six. Furthermore, we show that both problems admit polynomial-time algorithms on chordal graphs, bipartite graphs, and cographs. For each of these graph classes, we give a non-trivial upper bound on the number of Kempe-changes needed in order to certify that two k-colorings are Kempe-equivalent.
Supported partially by JSPS and MAEDI under the Japan-France Integrated Action Program (SAKURA). Also supported partially by the ANR Project GrR (ANR-18-CE40-0032) operated by the French National Research Agency (ANR), by JSPS KAKENHI Grant Numbers JP16H03118, JP16K16010, JP17K12636, JP17K19960, JP18H04091, JP18H05291, JP19J10042, JP19K11814, and JP19K20350, Japan, and by JST CREST Grant Numbers JPMJCR1401, JPMJCR1402, and JPMJCR18K3, Japan.
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Bonamy, M. et al. (2019). Diameter of Colorings Under Kempe Changes. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_5
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DOI: https://doi.org/10.1007/978-3-030-26176-4_5
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